Inflationary Cosmology in Scalar-Tensor Gravity: Reconstructing Higgs-Like Potentials
We study cosmology in scalar-tensor (Bergmann Wagoner) gravity, restricting the coupling function, to be constant. Rather than specify the form of the cosmological function, , the scalar field is modelled as a function which decays to its present value . Solutions of the field equations are found for which evolves from a large value (approximately 1) near the singularity to a small, non-zero value at later times, avoiding the problem of pre-inflationary collapse in standard general relativistic cosmology. Interpreting the model within the framework of Brans-Dicke theory with a scalar potential, we find that for suitable initial conditions, the reconstructed potential at late times for flat, open and closed universes is well described by a Higgs-like Mexican hat potential, quartic in , though this was not built in to the initial assumptions.
Programa de Pós-graduação em Física, Instituto de Física Armando Dias Tavares, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro – RJ.
Departamento de Física Teórica, Instituto de Física Armando Dias Tavares, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro – RJ.
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Keywords: scalar-tensor gravity, Brans-Dicke theory, Bergmann-Wagoner theory, running cosmological constant, inflation, Mexican hat potential.
For some time now it has been recognised that “old inflation” , in which inflation is generated by a scalar field, suffers from the shortcoming that inflation begins some time distant from the initial singularity, restricting the initial conditions of the Universe to those for which collapse does not occur before the phase transition that generates inflation. However Linde’s chaotic inflation  and “new inflation” models avoid this difficulty by producing inflation soon after the Big Bang. Many new inflation models assume that the potential for the scalar field has a particular form (see  for a recent review).
In this paper, we study cosmology in scalar-tensor gravity, also called Bergmann-Wagoner (BW) theory [4, 5] which naturally contains a variable cosmological function, - or running cosmological ‘constant’ - in addition to a coupling parameter, , which may also depend on the scalar field. In this theory, as in Brans Dicke theory, the scalar field is associated with the gravitational coupling such that .
We search for solutions that have various desirable properties: (1) “natural” initial conditions, in the sense that values of physical quantities near the singularity are of order 1 in Planck units; (2) sizable inflation is produced directly after the Big Bang; (3) they evolve to a final state with a small, but non-zero, cosmological “constant”; (4) the value of is compatible with observations; (5) they produce a variation in at late times which is compatible with present-day observations.
To this end we choose a model for , rather than a model for , simply demanding that it evolve from some initial almost constant value to a late-time almost constant value (which will be unity in Planck units). We find that a relatively small change in , of some is sufficient to produce solutions with the above properties. When we reconstruct the dependence of and the equivalent potential , we find that, at low energies, the latter is very well described by a Mexican hat/Higgs potential, with the scalar field evolving towards a local maximum at .
2 Scalar Tensor Gravity
The scalar-tensor field equations for BW theory are derived from the variational principle applied to the action 
where is the coupling parameter, the cosmological function and describes the non-gravitational part of the action (throughout the paper we use reduced Planck units with . As with Brans-Dicke (BD) theory, the gravitational constant of General Relativity is replaced by a function , where is a scalar field, with the BD cosmological constant and coupling constant generalized to functions and respectively. Using the subscript to denote present-day values, and assuming homogeneity, in Planck units . Observational evidence [7, 8, 9] suggests that today , or in Planck units .
BW theory generalizes BD theory, for which the gravitational part of the action may be written
where the coupling parameter is now constant, and .
If a scalar potential is added to the BD action then the gravitational action becomes 
and we see that BW theory with constant is equivalent to BD theory plus a scalar potential with the identification
Though the theories are equivalent, we prefer the BW approach with the interpretation of a time-varying cosmological function (or running cosmological constant) as it lends itself more clearly to our analysis, as well as leaving open the possibility of studying the effect of a variable .
The field equations derived from (1) are
We note that, different from pure BD theory, where implies , effectively demanding a vacuum or radiation fluid, there is no such restriction in BW theory if .
The standard approach in scalar-tensor cosmology is to specify the potential , or equivalently , and study the subsequent evolution of the variables in the model. Here we adopt a different procedure: we model the evolution of as a function that decays from an initial value, to its present-day value , hoping to identify solutions that generate inflation from , with evolving to a smaller, constant value as . Once these solutions are identified we reconstruct the functions and .
We assume that the energy-momentum tensor is that of a perfect fluid
with the energy density, the pressure of matter and the four-velocity with .
Supposing a homogeneous and isotropic universe, all variables depend only on time, , and the space-time is described by the Friedmann Lemaître Robertson Walker (FLRW) metric in the form
Since we are mainly interested in evolution close to the Big Bang, we use the equation of state . To reduce the amount of freedom in the model, we impose constant (hence making the model equivalent to BD plus scalar potential). Using an overdot to denote a time derivative the field equations for and are, respectively,
Considering as equation (6) may be rewritten
Finally, the Bianchi identity yields
Written in terms of this final equation is just the usual energy-momentum conservation equation with solution .
We are searching for solutions where decays from a large initial value to a much smaller value, with two-tier inflation. This suggests a model in which evolves from an initial value near the singularity (which we take as rather than ) to its present value of , with approximately constant for long periods around these values. Between these two eras, the intermediate decay of , occurring around say, could be caused by some phase transition at the corresponding energy.
A model for with such a smooth behaviour is
where governs the rate of decay. For this function (and consequently ) is approximately constant around , decaying smoothly around , and tending asymptotically to . Since tends to a constant value for , will also tend to a constant, in general different from .
This cosmology thus depends on the four parameters , , and , as well as the values of , and at . Siince is assumed constant, we can use present-day limits, the most stringent of which are given by the analysis of signals from the Cassini spacecraft . These limit the PPN parameter . From the relation 
we have that, to two standard errors, or .
Some articles on scalar-tensor cosmology [11, 12] use values of which are negative with small modulus (typically around ), incompatible with the Cassini observations. However fitting of type I Supernova data  suggests that Brans-Dicke theory with describes these observations on cosmological scales slightly better than the CDM models. This may be taken as evidence for a time-varying or spatially dependent , both of which fall within the scope of BW theory. Here, however, we maintain constant and positive.
We integrate the equations from the Planck time with all initial values close to unity. Our aim is to determine whether there exist initial conditions which, for the proposed model for , produce a cosmology with inflation from , while decays to a smaller value compatible with an observed value of the cosmological “constant” today. Evidently it is unrealistic to integrate the Planck times until the present day, but we can study what occurs for a reasonable number of Planck times around the period during which decays, for models with .
For the flat, open and closed cases we model by (16) with the same values of , and . This latter value produces a decay period (calculated as 95% of the fall) that lasts approximately . The time in the middle of the decay, , is chosen so that for a substantial time before it decays. We choose in all cases, compatible with the limits stated in the previous section. The equations were integrated using an adaptive Runge-Kutta 8th/9th order method due to Verner  implemented by the authors using the multiple precision MPFR library  with 1000 bits (300 decimal digits), and an error per step of .
To check that the numerical integrator was producing the desired precision, we evaluated using (13). For example the value of for the flat model studied below is shown in Figure 1, confirming that the error is around , in line with expectations.
3.1 Flat Universe
The first graph shows the evolution of from until . We see that, during this period, decays from approximately unity to . Integration up to does not significantly change this value, which is evidently much higher than today’s observed value of , but it does show that in a short time can decay significantly from a value of 1 near the singularity to a much smaller value. As we shall see shortly, other values of the parameters in the potential and initial conditions, the decay in can be greater.
The second graph shows the variation of the Hubble parameter during the same period. We notice that also decays, but in two distinct steps, falling from 1 to 0.6 in just one Planck time (seen in detail in the third plot), followed by a period when is approximately constant (a first inflationary period) up to . After this, decays abruptly (together with ), reaching around and changing little after that: at , . In this final stage, the Universe can be described as passing through a second, slower inflationary phase. During both inflationary phases and we have (approximate) de-Sitter solutions. With these initial conditions, the scale factor, , undergoes 400 e-folds during the first inflationary phase, thereafter increasing at a much slower rate.
Hence, albeit on a different time scale, we have the main characteristics needed to describe a Universe in which an initial inflationary period helps solve the horizon and flatness problems, while at late times a residual cosmological ‘constant’ produces an accelerated expansion. Inflation appears very soon after the universe emerges from the quantum era, thereby problems of early recollapse are avoided; and while, in General Relativity the primordial inflaton and late-time cosmological constant are normally considered separate entities, in BW theory they are naturally connected within the model.
It is interesting to ask what type of potential in BD theory would produce this type of solution. In figure 3 we reconstruct from and the equivalent BD potential (4). The graph on the right shows in detail the low-energy behaviour of the potential, which has a minimum away from . This low-energy potential is exceptionally well described by a one-dimensional Mexican Hat function around of the form
with , and . The fit was performed by Maple with 204 data points in the interval . We note that the ratio while, for the Higgs potential this ratio is or . Different choices of the parameters in the model for produce potentials with different values of , and . For example, since and at late times, a small late-time value of is equivalent to a small value of .
It is not difficult to obtain smaller values of at late times. For example, by slightly altering the decay rate parameter in the potential to , (for which ) we find that falls by a factor of to , as plotted in the first two graphs of figure 4. We not there is a period between and when is negative, but this does not lead to a recollapse. Once again, for low energies () the potential has the form (17) with , and . It is interesting that evolves towards a local maximum of the potential which is, in principle, an unstable equilibrium point. However tends to this maximum only as so this instability is effectively irrelevant.
3.2 Closed Universe
For the same parameters were used in the model of with the initial conditions slightly modified to satisfy the field equations with , giving for the same value of . Figure 5 shows the results of the integration. The only significant difference in the results is that the initial fall of reaches a minimum around before recovering to a value of around 0.6. The potential is once again well described by the model (17) with the same values of the parameters. The evolution of (not shown) again undergoes 400 e-folds during the primordial inflationary phase.
3.3 Open Universe
Exactly the same model for was used in the analysis for , but with different initial conditions: , , and consequently . In figure 6 we see that the results are initially very similar to the flat model, but there is a larger final value of , and consequently larger values of and a faster late-time expansion of , though again the variation of from the Big Bang until the end of the decay in is very similar to the cases.
The intention of this work is to study cosmological models within a restricted version of the Bergmann-Wagoner theory of gravity for which the coupling parameter is constant. This restriction makes the theory equivalent to Brans-Dicke theory with a scalar potential. A simple model for the scalar field was proposed in which is approximately constant in the early and late stages, passing smoothly through a period when it rapidly decays. For open, closed and flat FLRW models, solutions were sought and found in which the cosmological function, evolves from a value close to unity (in Planck units) near the Big Bang to a smaller almost constant value at late times. Because of computational restrictions, the evolution was studied over ‘short’ time scales (in cosmological terms) but we believe that this behaviour can be reproduced over longer time scales by appropriate choices of the parameters and initial conditions.
It was hoped that the problem of the fine tuning of initial conditions which beset standard inflation would be solved with a natural initial value of driving primordial inflation to avoid an early collapse. In the desired scenario, would subsequently decay naturally to a value close to, but not exactly, zero, compatible with present-day observations. On the positive side, solutions with these characteristics have been determined. However different choices of initial conditions and/or parameters in the scalar field model can produce other sorts of behaviour, such as early collapse.
Though our approach was initially couched in terms of a running cosmological function , within Bergmann-Wagoner theory, an alternative point of view in which is substituted by an equivalent potential within Brans-Dicke theory was also investigated. Curiously, it was found that the equivalent potential at low energies has a form of a Higgs-like or Mexican hat potential, centred around , with the BD scalar field evolving towards the local maximum there. In some sense this is to be expected from our model of : since we demand that tends towards as , the late-time potential must be . What is unexpected, and certainly not built into the assumptions, is the form of at low energies and the fact that corresponds to a local maximum of . It is interesting to speculate whether or not other forms of with similar early and late-time behaviours are associated with similar potentials.
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